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January  2020, 40(1): 33-46. doi: 10.3934/dcds.2020002

Hereditarily non uniformly perfect non-autonomous Julia sets

Mark Comerford 1, , Rich Stankewitz 2, and  Hiroki Sumi 3,

1. 

Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, USA
2. 

Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA
3. 

Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan

Received  October 2018 Revised  June 2019 Published  October 2019

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $ 1 $ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [15].

Keywords: Hereditarily non uniformly perfect sets, non-autonomous iteration.
Mathematics Subject Classification: Primary: 30D05; Secondary: 28A80.
Citation: Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002
References:
[1]

L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.  Google Scholar

[2]

Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.  doi: 10.21042/AMNS.2016.2.00034.  Google Scholar

[3]

E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[4]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[5]

M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.  Google Scholar

[6]

M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.  Google Scholar

[7]

A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. Google Scholar

[8]

K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.  Google Scholar

[9]

J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.  Google Scholar

[10]

A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.  doi: 10.1017/S0305004100076192.  Google Scholar

[11]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.   Google Scholar

[12]

R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.  doi: 10.1090/S0002-9939-1992-1106180-2.  Google Scholar

[13] C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.   Google Scholar
[14]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar

[15]

L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.  doi: 10.1090/tran/6490.  Google Scholar

[16]

O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.   Google Scholar

[17]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar

[18]

R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583.  Google Scholar

[19]

R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.  doi: 10.3934/dcdss.2019150.  Google Scholar

[20]

H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar

[21]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.  doi: 10.1017/S0143385701001286.  Google Scholar

[22]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532.  Google Scholar

[23]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.  doi: 10.1017/S0143385709000923.  Google Scholar

[24]

W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.  doi: 10.1007/BF02901155.  Google Scholar

show all references

References:
[1]

L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.  Google Scholar

[2]

Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.  doi: 10.21042/AMNS.2016.2.00034.  Google Scholar

[3]

E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[4]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[5]

M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.  Google Scholar

[6]

M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.  Google Scholar

[7]

A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. Google Scholar

[8]

K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.  Google Scholar

[9]

J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.  Google Scholar

[10]

A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.  doi: 10.1017/S0305004100076192.  Google Scholar

[11]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.   Google Scholar

[12]

R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.  doi: 10.1090/S0002-9939-1992-1106180-2.  Google Scholar

[13] C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.   Google Scholar
[14]

C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar

[15]

L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.  doi: 10.1090/tran/6490.  Google Scholar

[16]

O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.   Google Scholar

[17]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar

[18]

R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583.  Google Scholar

[19]

R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.  doi: 10.3934/dcdss.2019150.  Google Scholar

[20]

H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar

[21]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.  doi: 10.1017/S0143385701001286.  Google Scholar

[22]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532.  Google Scholar

[23]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.  doi: 10.1017/S0143385709000923.  Google Scholar

[24]

W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.  doi: 10.1007/BF02901155.  Google Scholar

Figure 1.  How the survival sets $ {\mathcal S}_k $ are nested. The pictures show preimages of $ \overline {\mathrm D}(0,2) $ at stages $ M_k $ (in red) and $ M_{k-1} $ (in blue) with $ m_k = 3 $. The dashed blue circle is $ {\mathrm C}(0,2) $ while the unit circle is shown in black. Observe how $ Q_{M_{k-1},M_k}^{-1}(\overline {\mathrm D}(0,2)) \subset \overline {\mathrm D}(0, 2) \setminus \overline {\mathrm D}(0, 1) $ as in Remark 1(c) is shown in red at Stage $ M_{k-1} $
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Figure 2.  Schematic for the proof of Theorem 1.6 in the case where $ \limsup |c_k| = +\infty $. Note how the round annulus $ {\mathrm A}(\sqrt{-c_{k}}, 1, \sqrt{|c_{k}|}) $ at stage $ M_{k-1} + m_k $ (in this case $ M_1+m_2 $) is pulled back conformally first by the preimage branches of $ Q_{M_{k-1},M_{k-1}+m_k} $ to form half the members of the collection $ \mathcal C $ at Stage $ M_1 $. Then the preimage branches of $ Q_{M_{k-1}} $ pull back the annuli in $ \mathcal C $ (one of which is visible in the zoomed box) to conformal annuli which separate the components of $ \mathcal{S}_k $ at stage $ 0 $
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